The class blog for Math 3010, fall 2014, at the University of Utah

Category Archives: Math and Nature

Who knew that an unlikely friendship and a few games of cricket with one of the greatest mathematicians in the early 20th Century could lead to a breakthrough in population genetics?

Today, it is almost commonplace for us in the scientific community to accept the influence natural selection and Mendelian genetics have on one another, however for the majority of human history this was not the case. Up until the early 1900s, many scientists believed that these concepts were nothing more than two opposing and unassociated positions on heredity. Scientists were torn between a theory of inheritance (a.k.a. Mendelian genetics) and a theory of evolution through natural selection. Although natural selection could account for variation, which inheritance could not, it offered no real explanation on how traits were passed on to the next generation. For the most part, scientists could not see how well Mendel’s theory of inheritance worked with Darwin’s theory of evolution because they did not have a way to quantify the relationship. It was not until the introduction of the theorem of genetic equilibrium that biologists acquired the necessary mathematical rigor to show how inheritance and natural selection interacted. One of the men who helped provide this framework was G.H. Hardy.

G. H. Hardy. Image: public domain, via Wikimedia Commons.

Godfrey Harold (G.H.) Hardy was a renowned English mathematician who lived between 1877-1947 and is best known for his accomplishments in number theory and for his work with the another great mathematician, Srinivasa Ramanujan. For a man who was such an outspoken supporter of pure mathematics and abhorred any practical application of his work[5], it is ironic that he should have such a powerful influence on a field of applied mathematics and help shape our very understanding of population genetics.

How did a pure mathematician come to work on population genetics? Well it all started with a few games of cricket. Whilst teaching at the University of Cambridge, Hardy would often interact with professors in other departments through friendly games of cricket and evening common meals [1]. It was through these interactions that Hardy came to know Reginald Punnett, cofounder of the genetics department at Cambridge and developer of Punnett Squares, which are named for him, and developed a close friendship with him[13].

Punnett, being one of the foremost experts in population genetics, was in the thick of the debate over inheritance vs. evolution. His interactions with contemporaries like G. Udny Yule, made him wonder why a population’s genotype, or the genes found in each person, did not eventually contain only variations, known as alleles, of a particular gene that are dominant. This was the question he posed to Hardy in 1908, and Hardy’s response was nigh on brilliant. The answer was so simple that it almost seemed obvious. Hardy even expressed that “I should have expected the very simple point which I wish to make to have been familiar to biologists’’ [4]. His solution was so simple in fact that unbeknownst to him, another scientist had reached the same conclusion around the same time in Germany [17]. In time, this concept would be known as Hardy-Weinberg Equilibrium (HWE).

In short, HWE asserts that when a population is not experiencing any genetic changes that would cause it to evolve, such as genetic drift, gene flow, selective mating, etc., then the allele (af) and genotypic frequencies (gf) will remain constant within a given population (P’). To calculate the gf for humans, a diploid species that receives two complete sets of chromosomes from their parents, we simply look at the proportion of genotypes in P’.

0 < gf < 1

To calculate the af, we look at the case where either the gene variation is homozygous and contains two copies of the alleles (dominant—AA || recessive—aa) or heterozygous and only has one copy of each allele (Aa). P’ achieves “equilibrium” when these frequencies do not change.

Hardy’s proof of these constant frequencies for humans, a diploid species that receives two complete sets of chromosomes from its parents, is as follows[1][4]:

If parental genotypic proportions are p AA: 2q Aa: r aa, then the offspring’s would be (p + q)2: 2(p + q)(q + r): (q + r)2. With four equations (the three genotype frequencies and p + 2q + r = 1) and three unknowns, there must be a relation among them. ‘‘It is easy to see that . . . this is q2 = pr”

Which is then broken down as:

q =(p + q)(q + r) = q(p + r) + pr + q2

Then to:

q2 = q(1- p – r) – pr = 2q2 – pr ——-> q2 = pr

In order to fully account for the population, the gfand af must sum to 1. And, since each subsequent generation will have the same number of genes, the frequencies remain constant and follows either a binomial or multinomial distribution.

One important thing to keep in mind, however, is that almost every population is experiencing some form of evolutionary change. So, while HWE shows that the frequencies don’t change or disappear, it is best used as a baseline model to test for changes or equilibrium.

When using the Hardy-Weinberg theorem to test for equilibrium, researchers divide the genotypic expressions into two homozygous events: HHο and hhο. The union of each event’s frequency ( f ), is then calculated to give the estimated number of alleles (Nf). In this case, the expression for HWE could read something like this:

Nf = f(HHο) ∪ f(hhο)

However, another way to view this expression is to represent the frequency of each homozygous event as single variable, i.e. p and q. Using pto represent the frequency of one dominant homozygous event (H) and qto represent the frequency of one recessive homozygous event (h), gives the following: p = f(H) and q = f(h). It then follows that p² = f(HHο) and q² = f(hhο). By using the Rule of Addition and Associative Property to calculate the union of the two event’s frequencies, we are left with F = (p+q)². Given that the genotype frequencies must sum to one, the prevailing expression for HWE emerges when F is expanded:

F = p² +2pq + q² = 1

Using this formula, researchers can create a baseline model of P’ and then identify evolutionary pressures by comparing any subsequent frequencies of alleles and genotypes (F∝) to F. The data can then be visually represented as a change of allele frequency with respect to time.

HWE represents the curious situation that populations experience when their allele frequencies change. This situation is realized by first assuming complete dominance, then calculating the frequency of alleles, and then using the resultant number as a baseline with which to compare any subsequent values. Although there are some limitations on how we can use HWE—namely, identifying complete dominance, the model is very useful in identifying any evolutionary pressures a population may be experiencing and is one of the most important principles in population genetics. Developed, in part, by G.H. Hardy, it connected two key theories: the theory of inheritance and the theory of evolution. Although, mathematically speaking, his observation/discovery was almost trivial, Hardy provided the mathematical rigor the field sorely needed in order to see that the genotypes didn’t completely disappear and, in turn, forever changed the way we view the fields of biology and genetics.

Pearson, Karl. “Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs.”Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character(1903): 1-66.

Pearson, K., 1904. Mathematical contributions to the theory of evolution. XII. On a generalised theory of alternative inheritance, with special reference to Mendel’s laws. Philos. Trans. R. Soc. A 203 53–86.

Most mathematically inclined people are familiar with the famous and unique Fibonacci sequence. Defined by the recurrence relation (*) Fn=Fn-1+Fn-2 with initial values F1=1 and F2=1 and (or sometimes F0=1 and F1=1), the Fibonacci sequence is an integer sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …) with many remarkable mathematical and real world applications. However, it seems that few are as well informed on the man behind this sequence as they are on the sequence itself. Did you know that Fibonacci didn’t even discover the sequence? Of course not! Predating Fibonacci by almost a century, the so called “Fibonacci sequence” was actually the brainchild of Indian mathematicians interested in poetic forms and meter who, through studying the unique arithmetic properties of certain linguistic sequences and syllable counts, derived a great deal of insight into some of the most fascinating mathematical patterns known today. But with a little bit of time (few hundred years), some historical distortion, inaccurate accreditation[1], and a healthy dose of blind western ethnocentrism and voila! Every high school kid in America now thinks there is a connection between Fibonacci and pizza. Or is it Pisa? (That’s a pun, laugh.) While often given more credit than deserved for the “discovery” of the sequence, Fibonacci was nonetheless an instrumental player in the development of arithmetic sequences, the spread of emerging new ideas, and in the advancement of mathematics as a whole. We thus postpone discussion of Fibonacci’s sequence – don’t worry, we shall return – to examine some of the other significant and often overlooked contributions of the “greatest European mathematician of the middle ages.”[1]

Born around the year 1175 in Pisa, Italy, Leonardo of Pisa (more commonly known as Fibonacci) would have been 840 years old this year! (Can you guess the two indexing numbers between which Fibonacci’s age falls?[2]) The son of a customs officer, Fibonacci was raised in a North African education system under the influence of the Moors.[3] Fibonacci’s fortunate upbringing and educational experience allowed him the opportunity to visit many different places along the Mediterranean coast. It is during these travels that historians believe Fibonacci may have first developed an interest in mathematics and at some point come into contact with alternative arithmetic systems. Among these was the Hindu-Arabic number system – the positional number system most commonly used in mathematics today. It appears that we owe a great deal of respect to Fibonacci for, prior to introducing the Hindu-Arabic system to Europe, the predominant number system relied on the far more cumbersome use of roman numerals. It is interesting to note that while the Hindu-Arabic system may have been introduced to Europe as early as the 10th century in the book Codex Vigilanus, it was Fibonacci who, in conjunction with the invention of printing in 1482, helped to gain support for the new system. In his book Liber abbaci[4], Fibonacci explains how arithmetic operations (i.e., addition, subtraction, multiplication, and division) are to be carried out and the advantages that come with the adoption of such a system.

Whereas the number system most familiar to us uses the relative position of numbers next to each other to represent variable quantities (i.e., the 1’s, 10’s, 100’s, 1000’s, … place), Roman numerals rely on a set of standard measurement symbols which, in combination with others, can be used to express any desired quantity. The obvious problem with this approach is that it severely limits the numbers that can be reasonably represented by the given set of symbols. For example, the concise representation of the number four hundred seventy eight in the Hindu-Arabic system is simply 478 in which “4” is in the hundreds place, “7” is in the tens place, and “8” is in the ones place. In the Roman numeral system, however, this same number takes on the form CDLXXVIII. As numbers increase arbitrarily so does the complexity of their Roman numeral representation. The adoption of the Hindu-Arabic number system was, in large part, the result of Fibonacci’s publications and public support for this new way of thinking. Can you imagine trying to do modern mathematical analysis with numbers as clunky as MMMDCCXXXVIII??? Me either. Thanks, Fibonacci!

Fibonacci’s other works include publications on surveying techniques, area and volume measurement, Diophantine equations, commercial bookkeeping, and various contributions to geometry.[4] But among these works nothing stands out more than that of Fibonacci’s sequence – yes, we have returned! Among the more interesting mathematical properties of Fibonacci’s sequence is undoubtedly its connection to the golden ratio (shall be defined shortly). To illustrate, we look momentarily at the ratios of several successive Fibonacci numbers. Beginning with F1=1 and F2=1 we see that the ratio F2/F1=1. Continuing in this manner using the recurrence relation (*) from above or any suitable Fibonacci table we find that F3/F2=2, F4/F3=3/2, F5/F4=5/3,F6/F5=8/5, F7/F6=13/8, F8/F7=21/13, … As the indexing number tends to infinity, the ratio of successive terms converge to the value 1.6180339887… (the golden ratio) denoted by the Greek letter phi. We may thus concisely represent this convergent value by the expression as the lim n–> infinity (Fn+1/Fn). Studied extensively, the golden ratio is a special value appearing in many areas of mathematics and in everyday life. Intimately connected to the concept of proportion, the golden ratio (sometimes called the golden proportion) is often viewed as the optimal aesthetic proportion of measurable quantities making it an important feature in fields including architecture, finance, geometry, and music. Perhaps surprisingly, the golden ratio has even been documented in nature with pine cones, shells, trees, ferns, crystal structures, and more all appearing to have physical properties related to the value of (e.g., the arrangement of branches around the stems of certain plants seem to follow the Fibonacci pattern). While an interesting number no doubt, we must not forget that mathematics is the business of patterns and all too often we draw conclusions and make big picture claims that are less supported by evidence and facts than we may believe. There is, in fact, a lot of “woo” behind the golden ratio and the informed reader is encouraged to be weary of unsubstantiated claims and grandiose connections to the universe. It is also worth mentioning that, using relatively basic linear algebra techniques, it is possible to derive a closed-form solution of the n-th Fibonacci number.

Figure 3-Computing the 18th Fibonacci Number in Mathematica.

Omitting the details (see link for thorough derivation), the n-th Fibonacci number may be computed directly using the formula Fn=((φ)(n+1)+((-1)(n-1)/(φ)^(n-1))/((φ2)+1).[5] While initially clunky in appearance, this formula is incredibly useful in determining any desired Fibonacci number as a function of the indexing value n. For example, the 18-th Fibonacci number may be calculated using F18=((φ)(18+1)+((-1)(18-1)/(φ)^(18-1))/((φ2)+1)=2584. Comparing this value to a list of Fibonacci numbers and to a Mathematica calculation (see picture above), we see that the 18-th Fibonacci number is, indeed, 2584. Without having to determine all previous numbers in the sequence, the above formula allows us to calculate directly any desired value in the sequence saving substantial amounts of time and processing power.

From the study of syllables and poetic forms in 12th-century India to a closed-form solution for the n-th Fibonacci number via modern linear algebra techniques, our understanding of sequences and the important mathematical properties they possess is continuing to grow. Future study may reveal even greater mathematical truths whose applications we cannot yet conceive. It is thus the beauty of mathematics and the excitement of discovery that push us onward, compel us to dig deeper, and to learn more from the world we inhabit. Who knows, you might even be the next Leonardo of Pizza – errrrr Pisa. What patterns will you find?[1] French mathematician Edouard Lucas (1842-1891) was the first to attribute Fibonacci’s name to the sequence. After which point little is ever mentioned of the Indian mathematicians who laid the groundwork for Fibonacci’s research.

A few months ago I was sitting at home watching one of those shows about the universe, you know where they try to condense everything there is to know about our world into a few short episodes? This particular episode was about Isaac Newton and all of his work. In this episode they were discussing how he invented Calculus and how he forever changed the way that we understand our universe. At this point I was pretty intrigued when my boyfriend raised a question I never really gave much thought to. He said to me “Do you believe math is something we discovered or something we invented?” My immediate reaction was that math was a discovery; there is no way that we just made all of this up! After this conversation occurred I started to notice that this was a question I began to think about often, but I never really could come up with a solid answer. So I will raise the same question again, was Math invented or discovered?

Fibonacci Sequence in a sunflower. Image:Ginette, via Flickr.

Let’s start with the discovery side of things; there are many different mathematicians who believe that math was a discovery, such as Plato and Euclid. Mathematical Platonism is “the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.”[1] This philosophical viewpoint is stating that our universe is made up entirely of math. When we begin to understand math we are allowing ourselves to understand more about how the world around us works [2]. Have you ever thought about how math occurs in nature, that there are patterns and sequences all around us? Euclid believed that nature was a physical manifestation of math [3]. Examples of mathematics in nature include honeycombs, wings of insects, shells, and flowers. We also find the opposite of patterns in nature, uniqueness. The theory that no two snowflakes are the same is an example of uniqueness occurring in nature. Another more modern theory that supports the notion that math is a discovery is the mathematical universe hypothesis, which was proposed by a cosmologist Max Tegmark. This theory states, “Our external physical reality is a mathematical structure.”[4] Basically he is saying that math is not necessarily used to describe our universe, but rather our universe is one mathematical object. I think this theory is very intriguing and would make perfect sense. It would explain why math can be applied to everything that we know.

On the other side we have the belief that math is an invention. The most common theory is that math is a completely human construct, which we made up in order to help us have a better understanding of the world around us. This theory is called the intuitionist theory. The theory is a rejection to Mathematical Platonism and states that “The truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition.”[5] Opposing the mathematical universe hypothesis is Gödel’s first incompleteness theorem. His theorem states that any theory that it has axioms can’t be consistent and complete at the same time. [6] This theory would show that math itself is like one giant loop. Every time we solve one problem based on assumptions we gain another problem that we must now base on assumptions we made from the last problem. This cycle will continue to repeat itself over and over and is inexhaustible.

Another common observation about math is how we actually carry out the process. If math were a discovery would we always have the same method for each problem. As shown in class the Egyptians had a completely different way to multiply that can be more effective than our current system of multiplication because it involves less memorization. Are our different methods for the same math problem enough to show that math is an invention? Or is it enough that we can get to the same solution, so the process isn’t as important? There is even the possibility that there are more discoveries to be made which could end our need for different methods to get to the same solution. There could be a missing link in our chain that we have to work around in order to get the solutions we need, but if we found that missing link we would only need one method to solve our mathematical problems.

In my own opinion the recurring theme of mathematics in nature is evidence enough for me to believe that math is a discovery and not an invention. With that said there are compelling arguments on both sides and it may take us years, if ever, to really prove whether or not math is a discovery or an invention

I’ve always wanted to travel to India, and I’m finally getting a chance to visit Chennai (along with some other places) this winter break. I’ll be teaching my company’s Chennai, India team about service oriented architecture automation – aka boring computer stuff. However, I’ve also set some time aside to go sightseeing on the company’s dime! We always seem to bring up India-birthed math topics, or mathematicians in class, so I thought it would be very fitting to blog about how India has impacted us! Make sure you get your Tetanus, Diphtheria, and Typhoid booster shots, this journey may get a little out of hand!

*Spoiler alert: You can’t contract any foreign diseases from a blog post.

When I think of India, computer software, call centers, spicy food, and the Taj Mahal come to mind. After making my way past these generalizations, I started to see how crucial this South Asian country’s mathematical contributions have been to mankind. India has been credited with giving the world many important mathematical discoveries and breakthroughs – place-value notation, zero, Verdic mathematics, and trigonometry are some of India’s more noteworthy contributions. This country has bred many game-changing mathematicians and astrologists. Over the course of my research I identified the “big three” mathematicians. The first, and arguably most important mathematician and astronomer (Ancient astronomers are similar to modern day astrologist!) in India’s history, was Aryabhata. Soon after Aryabhata, came Brahmagupta. Brahmagupta followed in Aryabhata’s footsteps and built upon some of his more groundbreaking theories. Nearly 500 years later Bhaskara II (Not to be confused with Bhaskara I.) was born. While building upon the mathematical and astronomical work of his forefathers, Bhaskara II also paved his own way to become one of the “greats”. The “big three’s” findings, laid down some of the most vital building blocks in the history of mathematics, but how has that impacted us?

We will start off on this journey with Aryabhata (sometimes referred to as Arjehir), a well-known astrologist and mathematician, born in the Indian city of Taregana sometime between 476-550 AD.He lived during a time period we now refer to as “India’s mathematical golden age” (400-600 AD), and it is of no surprise why historians recognize this time period; Aryabhata’s achievements really were golden. He is most noted for dramatically changing the course of mathematics and astronomy through many avenues, which he recorded in a variety of texts.

Sanskrit writing. Image: Diggleburnz, via Flickr.

Over the course of many wars and centuries, only one of Arybhata’s works survived. Aryabhatiya, which was written in Sanskrit at the age of 23, recorded the majority of his breakthroughs. Oddly enough, he only referenced himself 3 times throughout his work. Within this text, Aryabhata formulated accurate theories about our solar system and planets, all without a modern-day telescope. He recognized that there were 365 days in a year. He developed simplified rules for solving quadratic equations, and birthed trigonometry. Aryabhata’s original trigonometric signs were recorded as “jya, kojya, utkrama-jya and otkram jya” or sine, cosine, versine (equivalent to 1-cos(θ) ). He worked out the value of as well as the area of a triangle. Directly from Aryabhatiya he says: “ribhujasya phalashariram samadalakoti bhujardhasamvargah”. This translates to: “for a triangle, the result of a perpendicular with the half side is the area”. Most importantly, in my opinion, he created a place value system for numbers. Although in his time, he relied on the Sanskritic tradition of using letters of the alphabet to represent numbers. Aryabhata did not explicitly use a symbol for zero however. It kind of hard to conceptualize, but none of these things had ever been done, at least to this extent, before.

Brahmagupta

Brahmagupta. Image: public domain, via Wikimedia Commons.

Brahmagupta was born in Bhinmal, India presumably a short time after Aryabhata’s death in 598 AD. He wrote 4 books growing up, and his first widely accepted mathematical text was written in 624 when he was only 26 years old! I find it funny that most of the chapters in his texts were dedicated to disproving rival mathematicians’ theories. Brahmagupta’s most notable accomplishments were laying down the basic rules of arithmetic, specifically multiplication of positive, negative, and zero values. In chapter 7 of his book, Brahmasphutasiddhanta (Meaning – The Opening of the Universe),he outlines his groundbreaking arithmetical rules. In the context below, fortunes represent positive numbers, and debts represent negative numbers:

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

However it seems Brahmagupta made some mistakes when explaining the rules of zero division:

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Since our early teens we’ve know anything divided by zero is not zero. When zero is the denominator, the fraction will always “fall over” – that’s how I learned it as a youngin! However, we still have to give Brahmagupta credit, he was so close to getting it all right.

Bhaskara II

Bhaskara II is similar to the other mathematicians we’ve discussed in this post. He was born in 1114 AD, in modern day Karnataka, India. He is known as one of the leading mathematicians of India’s 12th century. He blessed the world with many texts but Siddhanta Shiromani, and Bijaganita (translates to “Algebra”) are the ones that have shined through the centuries. These specific texts documented some of his more important discoveries. In Bijaganita, Bhaskara demonstrated a proof of the Pythagorean theorem, and introduced a cyclic chakravala method for solving indeterminate quadratic equations:

y = ax2 + bx + c

Coincidentally, William Brouncker was credited for deriving a similar method to solve these equations in 1657, however his solution is more complex. From Siddhanta Shiromani, Bhaskara gave us these trigonometric identities:

If I had a dollar for every time I relied on these identities, or any of their variations throughout my mathematical career, I’d probably have enough money for a new laptop! Although Newton and Leibniz are credited for “inventing” calculus, Bhaskara had actually discovered differential calculus principles and some of their applications.

A World Without Aryabhata, Brahmagupta and Bhaskara II

I know this is a long shot, but let’s entertain the idea of a world without any of Aryabhata’s, Brahmagupta’s, orBhaskara’s work. Granted, future mathematicians would have undoubtedly discovered a portion of the “big three’s” breakthroughs, at least in one way or another. While it’s pretty obvious someone else would’ve invented a number system with a placeholder, or a zero equivalent, it’s not as clear with more complex things such as trigonometry. The foundation built by the “big three” could’ve altered slightly. This alteration could’ve given us a Leaning Tower of Pisa rather than an Eiffel tower – metaphorically speaking, that is. The main point you have to realize is: without the “big three” the progression of mathematics would have been slowed in one way or another, thus effecting our world today. If the “big three” didn’t exist there’s no telling how far back it could’ve set humanity.

That being said, these mathematicians’ theories, methods, and proofs served as building blocks for other mathematicians (globally). If you want to build out a brilliant theorem or proof, you have to start with, or at least incorporate the basics, at some point. Without these basics, the world would have been set back, at least in the realm trigonometry and algebra. It’s hard to imagine using any other number system than what we use today, especially without a numerical placeholder! Young children would be less eager to learn math because writing down large numbers would be a tedious process. What would we have used in place of zero? What about math with negative numbers?

Trigonometry electrifies our lives and rings in our ears. I think it is the biggest part of Aryabhata’s work that we take for granted. Without his trigonometric discoveries we wouldn’t have useful conventional electricity. The natural flow of alternating current, or AC current, is represented by the sine function. Electrical engineers and scientists use this function to model voltage and build the electronics we use every day. Alternating current primarily comes from power outlets, but it can also be synthesized in our electronic devices. Trigonometry is also extremely relevant today in music. Sine and cosine functions are used to visualize sound waves. This is especially important in music theory and sound production. A musical note or chord can be modeled with one or many sine waves. This allows sound engineers to morph voices and instruments into perfect harmony. However, Aryabhata is to blame for all that auto-tuned, T-Pain nonsense we hear on the radio! Lastly, trigonometry has a strong presence in modern day architecture. It’s a necessity when building complex structures and designs. We’d have to say goodbye to beautiful architecture and reliable suspension bridges if it weren’t for Aryabhata.

One can tell by the name itself what chaos theory might mean. Chaos means something that is unpredictable, random and unstable. There are many known and predictable phenomena in science such as electricity, gravity or chemical reactions; however, chaos theory examines things that are not possible to control. For example, nature: weather, earthquakes, clouds, trees, tsunamis, and tornadoes. Other than nature, there are human-related unpredictable things, such as the stock market and our brain states. Chaos theory is a field of mathematics that deals with complex systems whose behavior is highly sensitive to the slightest changes in initial conditions. For example, someone clapping their hands could change the weather, so even the smallest alterations can have big consequences.

Chaos theory emerged around the second half of the 20th century. This is because chaos theory has complex systems and these systems contain many elements that move. For this reason, computers are needed to calculate all the different possibilities. How did chaos theory come to be? A man named Edward Lorenz, a meteorologist, created a weather model on his computer in 1960. This weather model consisted of an extensive array of complex equations to predict weather conditions. This model always gave different sequence of numbers that represented weather conditions. One day he became curious and ran his own tests to see what the outcome would be. After running a sequence, he started running the same sequence halfway through, re-entering the numbers the first sequence had given him at that point. The results were not what he was expecting; they were entirely different from his first outcomes. The second time, he entered numbers that were rounded to three instead of six digits (for example, .506 versus .506127). Since the difference between these numbers is not much, he expected the results to be only slightly varied. However, that small error gave completely different outcome. Form this he concluded that even the slightest differences in initial conditions makes prediction of past or future outcomes impossible.

Image: J.L.Westover.

There are many principles of Chaos. One of them is the butterfly effect, also described by Lorenz. It is said that even a small butterfly flapping its wings in America can create a hurricane in Japan; if the butterfly did not flap its wing at the “right” time in space then the hurricane would not have happened. Even the smallest behavior has a direct effect in the future. Another principle is unpredictability. Since it is not possible to know all the initial conditions of a complex system in adequate detail, we can’t possibly know the outcome of those. As explained above, even the smallest change in numbers can lead to a big errors in prediction; outcomes can be completely different from what is expected.

We can never know for certain when we might have a storm or tsunami until few days before it’s about to happen. Similarly to the weather, chaos is present in our daily life. For example, the bus you usually take was late and you decided to take another bus, and randomly you meet a person, and you both start talking, he makes an impression on you, you go on a date with him, fall in love, get married and grow old together. Now imagine that the person had a similar situation: he decided to take this bus rather than his usual bus and met you. What if he never got on that bus at the right time to meet you, and what if you had decided to wait for you usual bus? It is scary to think about how one small decision makes such a big difference in your life.

In class, we talked about how dimensions can be non-integer values. We were given some examples of fractals, shapes with non-integer dimensions, and we were able to calculate the dimensions of some fairly simple fractals. But what about more complex shapes, that can’t be easily doubled, quadrupled, or so on? How does one measure, arbitrarily, the dimension of any shape?

Well, one of the ways to do so is to find the Hausdorff Dimension of a set. This concept of measuring dimensions was developed by (big surprise) Felix Hausdorff back in 1918.

The key idea is this: a “circle” with dimension 1 (a line) has its length vary proportionally to its “radius”. A circle with dimension 2 has its area vary proportionally to its radius squared, and so on with spheres and volume. To extrapolate, a “circle” with any dimension p would have its “p-volume” vary proportionally to its radius to the power of p. So, if p were 2.5, a 2.5 dimensional circle would, when doubled, increase in its 2.5 dimensional volume by 22.5 = about 5.66.

Now, hold onto your hats (if you don’t have a hat, go get one real quick, then hold onto it) because here’s where things get interesting. Let’s say you wanted to cover a 3-dimensional object with a bunch of smaller spheres, and measure the “4 dimensional volume” of the spheres. Well, we know that the 4-dimensional volume of an object is proportional to its radius4, so we can get an idea of how the 4-dimensional volume changes by simply adding together the radii4 of every sphere covering the object.

Now let’s take an arbitrary covering of our object by spheres and measure its 4-dimensional volume as above. We will get some number A. Now, how does that number change as our spheres shrink? Let’s say we, for example, replaced each sphere with 8 spheres, each having half the radius, and still managed to cover the object. Note that while the 3-dimensional volume of our spheres has remained the same (8 spheres of radius 1/2 have the same volume as 1 of radius 1), our 4-dimensional volume has been cut in half!

The main idea is that by shrinking the radii of our spheres, we can arbitrary decrease the 4-dimensional volume of the spheres covering our object. Since we can cover our object with spheres having arbitrary small 4-dimensional volume, it would make sense that the object would have 0 4-dimensional volume, which is consistent with how one would think about 4-dimensional area.

The key here is that this is true for any d-dimensional area if d is greater than 3, because our object is 3-dimensional. Thus, if we took the infimum of values for d such that this is true, we would get 3.

To reiterate, we had a 3-dimensional object. We were able to determine it was 3-dimensional because for any dimension d higher than 3, we could cover our object with spheres such that their d-dimensional volume was arbitrarily small. To be very precise:

The Hausdorff content of dimension d of an object is the infimum of numbers δ ≥ 0 such that there is some cover of the object by balls of radius r1, r2,… such that (r1d+r2d+r3d+…)< δ.

The Hausdorfff dimension of an object is the infimum of numbers d such that the hausdorff content of dimension d of an object is equal to 0.

Now, this approach matches quite nicely with our definition of integer dimensions, and provides a very nice way for us to expand that notion into other, non-integer dimensions. For example, this approach can actually be used in a surprising way: to find the dimension of coastlines.

Perhaps you’ve heard of the coastline paradox: that the more precise you try to measure the coastline, the longer it gets, seemingly without bound. This should signal to us that the coastline behaves as a fractal, and since the above method gives us a way to measure the dimension of arbitrary objects, we can use it to try and measure the dimension of the coastline.

Measuring the dimension of the coast of Great Britain. Image: Prokofiev, via Wikimedia Commons.

Here’s how it’s done: first, cover the coastline in large circles and measure the sum of the radii all put to some power p. Then, shrink the circles and measure again the sum of the radii all put to the power p. If this number keeps getting smaller, you’ve overestimated the dimension of the coast. If the number keeps getting bigger, you’ve underestimated it. If you keep doing this, you can fine-tune your estimate of the dimension. In fact, Mandelbrot did this very thing, and got that the coast of Great Britain had a fractional dimension of about 1.25, while the coastline of South Africa had a fractional dimension of about 1.02. While these are just estimates, it’s still cool to see how abstract ideas such as this can be used to measure things in real life.

We often hear people ask: “Why do we have to take math? We will never use it again.” The fact of the matter is math is all around, wherever we look. Even when you are camping up in the mountains, you can find something that is related to math. In the mountains, nature has made the golden ratio very prevalent in flowers and pinecones.

What is the golden ratio? On the website Live Science, the author Elaine J. Hom described it as the following: “The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.” What this is saying in an equation is a/b=(a+b)/a= 1.61803398874989…(ect.). This is also referred to the as phi, and it is an irrational number. This is how the ratio would be represented:

An important sequence is introduced when we are talking about the golden ratio. This sequence is called the Fibonacci sequence: 0,1,1,2,3,5,8,13… Each term is the sum in the two previous terms. The more you go to the right of the sequence the ratio of two terms right next to each other it will get closer to the Golden Ratio.

Now you might be asking yourself, “What does this all have to do with nature?” It has everything to do with nature. Let’s look at plants first. Usually, the number of leaves on the plant’s stem is arranged in a spiral pattern permitting the amount of sunlight the leaves need. The way the leaves or petals are arranged the Golden Ratio gives the ideal gap between the leaves or petals and they usually end up being a Fibonacci number. When we look at petals, we notice that they too have Fibonacci arrangements because when looking at them you will see a pattern. Each of these patterns you will see on petals of a flower all represent the Golden Ratio in their own way.

Looking at the pine cone you will notice the spiral that it naturally takes. Image: Böhringer Friedrich, via Wikimedia Commons.

Just like the petals and the leaves, pinecones are also in a spiral shape. Therefore, they too have Fibonacci qualities. They have two sets of spirals, one going in the clockwise direction and one going in counter clockwise direction, as you can see in the picture to the left. The numbers of spirals in the pinecones are almost always consecutive Fibonacci numbers. For example, there can be 8 spirals clockwise and 5 spirals counter clockwise. This shows the pinecones are related to the rational approximation of the Golden ratio (8/5).

Math is everywhere in our daily life. The Golden Ratio cannot only be seen in nature but it can be seen in everything around us. The Golden Ratio works hand in hand with the Fibonacci sequence. The more to the right we go in a Fibonacci sequence the more we can relate it back to the Golden Ratio. If we would just take a minute and look around, we will see that math is important and relevant in our daily lives.